In Ancient Greece, Aristarchus (310 BCE - 230 BCE) used a lunar eclipse to
measure the size of the moon relative to the size of the Earth. He found that
Size of moon ~ 1/3 Size of Earth
Using this plus Eratosthenes' measurement of the size of the Earth, Aristarchus
calculated that the distance to the moon is ~ 60 Earth radii.
Scale model of the solar system
In the following diagrams, all sizes and distances to scale, unless an object
is too small to be visible, in which case a green dot is used.
Distance to the sun
Aristarchus measured the distance to the sun by observing the moon when it is
half illuminated. In this configuration, the Earth-Moon-Sun system is at a
right angle and one measures the Moon-Earth-Sun angle.
If the sun were infinitely far away the angel would be 90 degrees,
and if the Sun is at a finite distance the angle is less than 90 degrees.
Aristarchus was unable to measure this angle precisely, but he was able to
show that it is at least 87 degrees. This established that the sun is
at least 20 times more distant than the moon.
Size of the sun
A solar eclipse tells us that the sun and moon have the same angular size.
Sun diameter = Moon diameter * Distance to sun / Distance to moon
Using this, Aristarchus estalished that the minimum size for the sun is
Sun diameter > 1/3 Earth diameters * 20
> 7 Earth diameters
This estalished that the sun is larger than the Earth, and was the first solid
clue for heliocentrism.
The measurement of the distance to the sun was refined by Aryabhata (476 CE -
550 CE).
Map of the world
Eratosthenes' map of the world
Ptolemy's map of the world
Ptolemy developed a system of latitude and longitude for mapping the world.
His map covered 1/4 of the globe and was the standard until the Renaissance.
Timeline
-310 -230 Aristarchus. Measures the distance to the sun.
-276 -195 Eratosthenes
-240 Eratosthenes measures the Earth's circumference to 20% error.
90 168 Ptolemy
476 550 Aryabhata. Refines the measurement of the distance to the sun.
1473 1543 Copernicus
1546 1601 Brahe
1600 Brahe measures accurate positions for the stars and planets over a period of 10 years.
1608 Lippershey invents the telescope
1609 Galileo builds a telescope
1609 Kepler uses Brahe's data to establish that planets orbit in ellipses.
1610 Galileo discovers the moons of Jupiter.
1639 Horrocks measures the transit of Venus, producing a value for the
distance to the sun that was low by a factor of 1/2.
1571 1630 Kepler
1564 1642 Galileo
1643 1727 Newton
1670 Picard measures the size of the Earth accurate to 1%.
1672 Richter and Cassini measure the parallax of Mars, which yielded a
value for the distance to the sun accurate to 10%.
1676 Romer uses the moons of Jupiter to measure the speed of light.
1729 Bradley measures the deflection of starlight due to the Earth's motion,
which gives a measurement for V/C, where V is the Earth's velocity.
1849 Fizeau makes the first measurement of the speed of light that doesn't
require astronomical data. 5% error.
1862 Focault measures the speed of light to .2% error.
1863 First measurements of stellar parallax and the distance to nearby stars.
2003 WMAP mission measures the Hubble constant to 5% precision.
Previous to this, the Hubble constant had an error of ~ 20%.
R = Earth-sun distance
V = Earth orbital velocity
T = Earth orbital time (1 year)
= 2 Pi R / V
t = Time for light to cross the Earth's orbit
= 2 R / C
C = Speed of light
Brahe's data consisted of measurements of angles between different objects.
This data could be used to establish the shape of orbits but not their size.
For example, if the size of the solar system were doubled along with the speeds of the
planets, the angles would stay the same and you wouldn't be able to tell the difference.
In 1639, Horrocks used a transit of Venus to measure the distance to the sun,
but this method is incapable of giving an accurate value, and it can only be
done once per century.
In 1672, Richter and Cassini measured the parallax of Mars which gives a result
for the distance to the sun that is more accurate than the Venus method. The
Mars method has an advantage over the Venus method in that it can be done once
every 26 months, when Mars is at closest approach.
In 1676, Romer used the moons of Jupiter to measure the time it takes for light to
cross the Earth's orbit. This gives a value for R/C.
In 1729, Bradley measured the deflection of starlight due to the Earth's motion,
which gives a measurement of V/C, or equivalently, a measurement of R/C.
In 1849, Fizeau produced the first measurement for the speed of light that was
independent of the Earth-sun distance R.
According to the Hubble law, distant galaxies recede from us with a speed of
Speed of galaxy = Hubble constant * Distance
Speed is easy to measure (from the redshift) and distance is difficult to measure.
Previous to 2003, the value of the Hubble constant was not well determined.
Parallax
Resolution of human vision 60 arcseconds
Resolution of a 10 cm telescope 1.0 arcseconds
Parallax of Alpha Centauri 1.4 arcseconds Nearest star
Parallax of 61 Cygni .31 arcseconds
Since the parallax of stars is undetectable to human vision, an astronomer in
Ancient Greece could conclude that the stars are vastly more distant than the sun.
First parallax measurements were done in 1838, when Bessel measured the parallax
of 61 Cygni, and Struve & Henderson measured the parallax of Alpha Centauri.
The limit of telescope resolution is 1 arcsecond due to air turbulence.
Solar system data
Moon radius = .273 Earth radii
Geosynchronus orbit = 6.6 Earth radii
Moon orbital radius = 60.4 Earth radii
Distance to sun = 23500 Earth radii
Jupiter radius = 10.9 Earth radii
Sun radius = 109 Earth radii
Moon orbital radius = .00257 A.U. (Astronomical unit = Earth orbit radius)
Distance to Earth L2 = .01 A.U.
Sun radius = .00465 A.U.
Moon orbital radius = .257 Earth Hill radii (Distance to Earth L2)
Earth-Venus distance = 28 Earth Hill radii
Earth-Mars distance = 52 Earth Hill radii
Earth-sun distance = 100 Earth Hill radii
Sun radius / Moon orbital radius = 1.81
1 Earth radius = 6371 km
1 A.U. = 1.50*10^11 meters
1 Earth mass = 5.974*10^24 kg
The distance to the Lagrange point L2 is the "Hill radius", an indicator of a
planet's gravitational influence.
If a moon is outside of 1/3 Hill radii, it will be stolen by the sun.
The moon is barely within this distance.
If two planets are within ~ 10 Hill radii of each other, their gravity will disrupt
each other's orbits.
